Soak the rice cakes in cold water for 20 minutes if they came refrigerated or frozen. Fresh, soft ones can go straight in.
Simmer 700 ml / 3 cups water with 6 dried anchovies and one 10 cm / 4 in square of dried kelp for 10 minutes, then strain out the solids.
Whisk the gochujang, gochugaru, soy sauce, sugar and garlic into the hot broth until no lumps of paste are left.
Add the drained rice cakes. Simmer over medium heat for 8 minutes, stirring every minute or so to keep them off the bottom.
Add the fish cake and the hard-boiled eggs. Simmer 4 more minutes.
Keep cooking until the sauce coats a rice cake and drips off slowly, about 2 to 3 minutes more.
Stir in the scallions, take the pan off the heat, and let it sit for 2 minutes before serving.